Why are there black edges on this textured mesh?
See the paint over I made - your edges are suffering from lack of pixels that are provided in the unwrap. It doesn't matter how big an image you use here if your model faces are only given a few pixels in that image.

If you were to rework that unwrap and allow more equal spacing for the faces in the green area compared to the pink area, you might see less of a problem. A way around this is to use a box mapped image texture to get a nice blended texture and then bake it through an emission shader to a blank image that uses the UV mapping, baking Combined. This way the pixels are arranged nicely on the model surface and then are mapped back to the uv mapping on a new texture to use that isn't showing signs of strain.